# Questions tagged [stochastic-differential-equations]

Stochastic differential equations (SDE) occur where a system described by differential equations is influenced by random noise. Stochastic differential equations are used in finance (interest rate, stock prices, …), biology (population, epidemics, …), physics (particles in fluids, thermal noise, …), and control and signal processing (controller, filtering, …).

115 questions
Filter by
Sorted by
Tagged with
3k views

• 131
474 views

### Integral of "white noise" multiplied by exponential term (multivariate Ornstein-Uhlenbeck)

Consider a matrix differential equation for the vector $\mathbf{y}$ of the form \begin{equation} \dot{\mathbf{y}}(t) = A \, \mathbf{y}(t) + \mathbf{b}(t) \, , \end{equation} where $A$ is a ...
• 1,943
1k views

### Using Itos Lemma To Derive an Ito Stochastic Differential Equation

So we have an Ito Stochastic Differential Equation with $b$ as a constant: $$dX_t = (bX_t +1)dt +2 \sqrt{X_t}dW_t$$ I then am told to let $Y_t = \sqrt{X_t}$ and thus derive the Ito stochastic ...
• 33
1 vote
124 views

### Determining Properties of Stochastic Differential Equation as Drift Tends to $0$

Consider the stochastic differential equation $$dS=\left(\mu S+\frac{\Lambda-S}{\omega}\right)dt+\sigma S dW_t.$$ such that $W_t$ is a Wiener process and $\mu,\sigma,\Lambda,\omega\in\mathbb{R}^+$. As ...
• 145
1 vote
125 views

• 1,694
714 views

### Definition of Stratonovich Integral

I have a doubt in definition of the Stratonovich integral. In "Stochastic Calculus for Finance" by Steven Shreve, he defines it using the midpoint $\frac {(t_i+t_{i+1})}{2}$ of the ...
2k views

• 12.2k
2k views

### How to solve system of stochastic differential equations?

I have the following two SDEs $$dN_1=(2a-1)pN_1dt+\alpha_1 N_1dW_1$$ $$dN_2=(2pN_1-\mu N_2)dt+\alpha_2 N_2dW_2$$ where $W$ is the standard Brownian motion/Wiener process. This isn't homework, I'm ...
• 123
6k views

### Expectation of stochastic differential equation

I have solved the nonlinear stochastic equation $dX_t=\frac{1}{2}a(a-1)X_t^{1-2/a}dt+aX_t^{1-1/a}dW_{t}$, by reducing it to a linear one (change of variables $Y_{t}=X_{t}^{1/a}$ and applying Ito ...
• 53
214 views

### Do there exist diffusions that do not solve any SDE?

Diffusions are continuous time stochastic processess having continuous paths and satisfying the strong Markov property. I know it is possible to characterize some diffusion processes as solutions to ...
• 364